Suppose $f$ is an entire function such that $f(D(0,R)) \subset D(0,M)$ for some $R,M > 0$.
Is it possible to prove that f must have at least one root in the open disk $D(0,R)$?
I really struggle at making any progress, my problem is if I knew any info about the behavior of $f$ on the boundary of $D(0,R)$ (for example if $f$ was constant there) I would be able to answer.
Can the question be answered as is? Any hints would be welcomed.